[/quote]
,
be a positive integer. There are two numbers written on a blackboard, one red and one blue. Initially, both are 0. We define the following procedure: at each step, we choose a nonnegative integer 
(not necessarily distinct from the previously chosen ones), and, if the red and blue numbers are 
and 
respectively, we replace them with 
and 
,
.
are positiv real number, such that 
prove inequality 
and 
.
be the set of all positive integers. Determine all subsets 
of 
such that
and 
intersect at 
and 
.
;
and 
,
and 
intersects 
.
on 
.
and 
meet at 
,
.
.
?
be a positive integer. A Japanese triangle consists of 
circles arranged in an equilateral triangular shape such that for each 
,
,
,
row contains exactly 
circles, exactly one of which is coloured red. A ninja path in a Japanese triangle is a sequence of 
,
,
and 
.
is prime, 
with 
and 
Y by Amir Hossein, Adventure10, Mango247
The quadrilateral 
is inscribed in a circle 
with the center 
. Its diagonals
intersect at
.
and 
are the centers of the 
and 
respectively.
Line through intersects at 
and 
. It also intersects 
at 
and 
respectively. and are lay inside of 
.
Prove that
.
is inscribed in a circle 
with the center 
. Its diagonalsintersect at
.
and 
are the centers of the 
and 
respectively.Line through
intersects at 
and 
. It also intersects 
at 
and 
respectively. and are lay inside of 
.Prove that
.Attachments:
is isosceles. Let 
and 
.
and 
. Also, 
and 
are collinear, (the same for 
).
,
.
is a parallelogram(by simple angle chasing) which means that 
and 
have the same circumradius,implying 
by sinus law,so 
.
